Transition between viscous dipolar and inertial multipolar dynamos

We investigate the transition from steady dipolar to reversing multipolar dynamos. The Earth has been argued to lie close to this transition, which could offer a scenario for geomagnetic reversals. We show that the transition between dipolar and multipolar dynamos is characterized by a three terms balance (as opposed to the usually assumed two terms balance), which involves the nongradient parts of inertial, viscous and Coriolis forces. We introduce from this equilibrium the sole parameter RoE(-1/3)ReE(2/3), which accurately describes the transition for a wide database of 132 fully three-dimensional direct numerical simulations of spherical rotating dynamos (courtesy of U. Christensen). This resolves earlier contradictions in the literature on the relevant two terms balance at the transition. Considering only a two terms balance between the nongradient part of the Coriolis force and of inertial forces provides the classical Ro/l(u). This transition can be equivalently described by Rel(u)(2), which corresponds to the two terms balance between the nongradient part of inertial forces and viscous forces.